Dekeyser, R.
1965
Physica 31
1405-1417
A CONVERGENT EXPRESSION
FOR THE PARTITION
FUNCTION OF A PARTIALLY
IONIZED HYDROGEN
PLASMA
by R. DEKEYSER*)
Instituut
voor Theoretische
Fysica,
Rijksuniversiteit,
Utrecht
synopsis
The partition function of a hydrogen plasma is derived, partly classically and
partly quantum-mechanically,
avoiding in this way at the same time the close distance
and the high orbit divergences. Our result is exact in the small density and infinite
temperature limit, but can be considered as a good approximation also for conditions,
where the ionization is incomplete. From this a Saha equation is derived, which
contains an effective lowering of the ionization potential. An expression is obtained for
the pressure, which is not simply the sum of the known pressures of the neutral and
ionized parts, but has a correction from the transitions between these parts.
Introdz&ion.
As is well known, the partition function of a hydrogen gas
diverges, due to the infinite number of energy levels, corresponding
to
high principal quantum numbers. Herzfeldr)
was the first to investigate
this problem, which he tried to solve by a cutoff at a certain maximum
Bohr radius. The interpretation of this maximum radius, however, was not
clear. Instead of this cutoff, Fowlers)
proposed a continuous decrease of
the occupation probabilities of the energy states, and Fermis)
derived a
concrete expression for this decrease. Variations on these methods were
presented by Beckerd),
Plancks),
Riewe
and Rompea).
Recently,
Larenz7)
gave a solution by a rather heuristic junction between the
weight functions for bound and free energy states.
An other problem is the non-existence of a classical hydrogen plasma:
there are always divergences from small electron-proton
distances, which
must be treated quantummechanically.
This difficulty is most often avoided by the assumption that the positive charges are smeared out. The
question arises, however, if a correct quantummechanical
treatment would
not reveal important corrections to the results obtained by the smearing out.
In the present paper we try to solve both problems at the same time.
The lowest bound energy stares are treated quantummechanically,
with
*) Aangesteld
navorser
van
het Belgisch
-
Nationaal
1405 -
Fonds
voor
Wetenschappelijk
Onderzoek.
1406
neglect
R. DEKEYSER
of the plasma
in which the atom is immersed.
The higher energy
states will be described classically and from a collective point of view. The
boundary between these two descriptions is still arbitrary (between certain
limits),
and we have to prove that our results do not depend on it.
In the first section we describe our system, and we investigate the conditions under which our assumptions
are valid. The second and third
sections
give the calculations
of the partition
function,
and in section 4 we
derive from it the Saha equation. In section 5 we prove that our expression
for the partition function is independent
of the definite choice for the
boundary between our two descriptions;
finally, as an application,
we
calculate the pressure of our gas.
5 1. General ideas. For definiteness we consider a plasma consisting of
N electrons and N protons, with masses m, and mP and charges -e and +e.
We introduce
M = me + mp,
m
=
memp
---,
M
We suppose that these particles are contained in a volume V, and denote
p = N/I/.
There will be for our particles a possibility to form bound pairs. It is
obvious to describe these pairs quantummechanically
by omitting the influence of the rest of the plasma; this must be possible if the plasma is
sufficiently
dilute. The possible internal energies of these pairs will then
approximately
be the energies E n of the hydrogen atom, corresponding to
the principal quantum number n with degeneracy ns. We do not take
account of the spin degeneracy.
We shall, however, only maintain this quantummechanical
description
up to a certain energy limit E, (which is negative). For every other electronproton pair, there still remain two possibilities.
If the distance r between
them is less than je2/E,I, their relative momentum p must be greater than
[2m(es/y + Ep)]*, in order to satisfy the condition that their internal energy
be greater than E,. If the distance is greater than le2/E,(, there is no lower
limit for that momentum.
Each of these possibilities
will be described
classically.
In order to be able to make such a cut between a classical and a quantummechanical description, there are some conditions to be fulfilled:
a.
The energy states above the limit must lie close enough together to
allow us to consider them as an “almost continuous spectrum”.
This
means
v> 1.
PARTITION
FUNCTION
OF A HYDROGEN
1407
PLASMA
b. The quantummechanical
energies of the hydrogen atom may not considerably be altered by the presence of the other plasma particles. This
will only be satisfied if the radius of the highest quantized orbit is much
smaller than the mean interparticle distance:
We impose one further
for convenience :
condition,
not so much for fundamental
reasons as
kBT > jEvI, where kB is Boltzmann’s constant and T the temperature.
This provides us with a small parameter p (&I @ = l/kBT) for series developments of otherwise hardly manageable functions:
The situation, which permits a perfect combination of conditions a and b
is the infinitely dilute plasma. If we want to generalize our calculations to
plasmas with densities up to 1016 cm-s, we shall have to content ourselves
with v N S-10. Condition c means that we have a high temperature description.
In the following table, we propose acceptable values of Y, corresponding
to several hydrogen plasma densities. In each case we indicate Tmin, the
minimum value of the temperature
that satisfies condition c.
c.
TABLE
Y
/
r,.(cm)
/
Tmin(‘K)
ro(cm)
1
possible
106
6.10-S
-
100
109
6.10-d
40
10’2
6.10-S
30
5.10-e
500
10’5
6. IO-6
10
5.10-T
5.103
10’8
6.10-T
5
10-1
2.104
p(cme3)
1
I
5. IO-5
10-S
50
.
300
We can also apply our results to low temperature
media as e.g. the
ionosphere by weakening condition b. We must then suppose that neutral
atoms do not disturb the electron orbits, so that we have to take account
only of the density of ionized particles in condition b.
To have a consistent description, we must further note that the quantummechanical form of the partition function of a system of t particles with
energy levels E and degeneracy g(E)
z g(E) exp(---BE)
corresponds
(1)
with the classical form
h-St/ . . . /drr
. . . drt dpr . . . dpt exp[-@E(rl,
. . . . rt; p1, . . . . pt)]
(2)
3 2. The different parts of the partition function. Let us suppose that we
have between our particles N, electron-proton
pairs in the n-th bound state
R. DEKEYSER
1408
with internal
greater
denote
energy En(n = 1, 2, . . . , v) and N’ pairs with a relative
than E, but with a relative
distance
less than
energy
Iez/E,I. We further
N”=N-N’-Nl-Nz-...-N,=N-NO.
(3)
The partition function of our plasma will then be the sum of the contributions from all possible divisions of our 2N particles in the N, and N’ pairs
and the 2N” free particles. Let us write it as follows:
z=
N! N!
x
;r
Nl! . . . NY! N'! N"! N"!
(N.),N’,N”
Z;“.Z’N’.Z”,
n=l
and we shall try to calculate each of the unknown factors. The factorials
arise from the number of ways in which each division can be performed.
We immediately note that a part of the summation can be carried out, so
that
Z=
Z,
N! N!
NO! N"!
NB+$=N
(2’ +
N"!
Zn)Avo’Z”*
;:
n=l
is very simple to write down: it contains
the degeneracy
92s of the
n-th hydrogen level, the probability factor exp(-_BE,)
and a factor
the translational
motion of the center of mass of the bound pair:
from
an2 exp(-DE,).
(5)
2’ contains the same factor from the center of mass motion, but we
describe the internal motion with the classical formula (2), taking account
of the boundary limits for the distance and relative momentum. Here too we
disregard the influence of the other plasma particles.
-Ey=
From now on, we write
A.
We then have
=
~(~~$jdE fdy““,t ’
e-BE
-A
=
1’
(?f?!-)‘.
A
m
-
4
s
A
dyy- 4 -!- l/y { B
A es-4 + $
[1 - @(~/B(Y -
41 e@y
I
PARTITION
FUNCTION
where G(x) is the error integral,
OF A HYDROGEN
1409
PLASMA
defined as
Q(X) = -d$fe-‘a
dt.
0
The difficult part in these calculations
We made use of the relation
is the integral over the error function.
00
s
dy (1 -
@( dp(y
-
A))] e&y = L
u
A
valid for j3 > a, which we integrated four times over a. Part of the remaining
calculations can be carried through exactly, the rest can be approximated
as a series in the small parameter A@. We do not give the lengthy calculations; the result is
2” can be written as an integral over the space and momentum variables
of the 2N” remaining particles. As there is no limit to be imposed on the
momentum
variables,
we can immediately
integrate
over them, which
yields a factor
On the space variables,
of the N” electrons
however, we have to impose the condition
may come closer than a distance
In a certain sense we have to describe the interaction
core (H.C.). The rest of 2” is thus
eG= f . . . f dxl . . . dxlvn dyi . . . dYN# exp[-/%V(xi,
H.C.
that none
ez/A to each proton.
potential
with a hard
. . . . XN”; yi, . . . . YN~)],
where we give coordinates x to the electrons and coordinates y to the protons.
The function W is the sum of N”s electron-proton interaction potentials
-e2/lxr
N”(N” -
1)/2 electron-electron
+e2/lx, -
-
YJL
and as many
ql
or
proton-proton
+e2/(Yt -
Yjrl*
interactions
1410
R. DEKEYSER
A useful trick for approximately
of es and to use
calculating
this, is to write 2es instead
1
1) = G(A = 0) + /dld$
G(il =
0
In this way, we get the result
where U(L) is the internal energy by the interaction strength 1e2, and can
be written with the correlation functions f+ and f- for particles of equal
and unequal charges, respectively :
u(a) =
s
v
dr F
N”2
[f+(r, ile2) -
f-(Y, nez)].
(8)
H.C.
We dropped terms of order N”-1. The correlation functions can be calculated
with the B.B.G.K.Y.
method, as has been done by Guernseys)
and
Shurea).
As a first approximation
to this, we can take
f+P>ae2)= exp[-PJe2/~l
f-Jr, ile2) = 0
1
for
f*(y,a@)
=
!F
1 F
y
<
e2,A
(9)
e-Kd\/Lr for
r > e2/A
with
With these approximations,
sq
we get the result
1
U(l)
= -
gN”Ke2 +
4nNN2e6
2vA2
[l + AB - CW2 log (49 + -..I.
0
More refined expressions
in the small parameters
(10)
of the correlation
N”e6
functions
<Y”>
El= -izG -
L-1 YO
and
Ke2
&2
=
~
A
=
2K(Yy>.
3
only contain corrections
PARTITION
FUNCTION
OF A HYDROGEN
PLASMA
1411
(By saying that a function is of order E, we shall mean hereafter that it does
not contain larger terms than linear ones in both ~1 and es).
In order to have a good connection between our classical and quantummechanical description, we would then be obliged to carry also the quantummechanical calculations to a higher order in these parameters. This would
mean :
1) to take account of the possibility
of two or more electrons being at the
same time in the immediate neighbourhood of the same proton;
2) to consider the screening effects of the plasma on the hydrogen energy
levels.
There has been done some work on this last topic by Miss Harris la),
and the resulting corrections on the pressure are of the same order of magnitude as our corrections (obtained in section 5). We doubt, however, if her assumption of a Debye-screened potential is correct up to lowest bound states.
In each case, this would lead us far out of the scope of our modest project,
and we prefer not to do it. Although ~1 and ~2 can be of the same order of
magnitude as Ap, we take our approximations for U to the second order in
A/l and to the lowest order in ~1 and ~2. In that approximation the above
result (10) is exact.
5 3. Result. Putting (5), (6), (7) and (10) in (4), we have for the partition
function of the plasma
where we abbreviated for the following
to first order in ~1 and ~2,
functions,
nN"e6
VAs
-W/V
\
@A49 + -11
[+
+
after developing
them
243 + 2(4)2 -
(12)
and
-v’+W
- 8(A/V WAB) + .
..I.
(13)
We remark at once that the N”-dependent part of F is much smaller than
the independent part, and that g is independent of NO. There are two terms
between the square brackets in the expression for g that cancel out against
1412
analogous
calculated
R. DEKEYSER
terms in the summation over the hydrogen levels, which can be
by partly substituting the summation by an integral, with
E,
= _
112e4
?GzK’
In this way g is of the form
g = g’ +
2nm
__
8122
@ 7ce6
x[Q
( >
+ 2&3 + 2(&3)2 -
&M3
log(M)
+
-..I (14)
where g’ is independent of A.
On the partition function (11) we apply the usual variational techniques
to get the largest term of the summation, and we drop the smaller terms.
We must look for a maximum, both in NO and N”, of the expression
)3(NO + N”) + log[(NO! N”! N”!)-1 g’V”FN”],
where ;Zis a Lagrange
= N. Differentiation
multiplier, to be chosen afterwards
of this expression yields
so that
A + log g = log NO
(15)
$, = log N”s.
i.+logF+$
NO + N”
(‘64
One immediately
verifies that the last term of the left hand side of (16a)
is of order E. Consequently, we can rewrite this last equation as
A + log
where the relation
+ s)
(‘6b)
with
as
s = O(E).
(17)
(15)-( 16) have the solution
NO=
The constant
log N”s,
between f and F can be written
f = F.(l
The equations
f=
ag; N” = l/q.
a = eA must be determined
so that
N=ag+dq=a(g+&).
(‘9)
Taking now only the largest term, and using (18)) (19) and Stirling’s
we have
=
N!
N”!
(18)
’j+,yN-N" - [(N"g + f,-($,"'"]".
formula
PARTITION
FUNCTION
OF A HYDROGEN
PLASMA
1413
Using (17)-( 19) one verifies directly that the square bracket can be written,
to first order in .s, as
(N”g + F)N>
where all terms containing
Z=
~“1
N!
N"N-N
”
A/3 cancel out. The result is
[v($fyJN b”g’+ v (Th;)*
(1 + &@P)]“,
(20)
which can easily be interpreted
as the product of a statistical
factor, a
proton part and an electron part. As the proton and electron masses are
not exactly equal to A4 and m, it is perhaps better to write
The second part in the square brackets represents the contribution
from
the free particles, whereas the first one represents the atoms. The factor N”
in that first part can be understood as follows: if we add a new electron to the
plasma, its probability to form an atom will be proportional to the number
N” of the free particles, with which it can form a bound state.
5 4. Saha-equation and effective ionizatiolz potential. The ratio between
the bound state part and the free particle part in (21) corresponds in a first
approximation
to the ratio between the numbers of neutral and ionized
atoms in the plasma, which is nothing else than the well known Saha
equationii).
To obtain this equation with the corrections in E, however, we
cannot start from (21), but we must solve the exact equations (18). The
corrections that appear can be interpreted as a lowering of the ionization
potential. It is too obvious, however, that the corrections, which contain
the arbitrary chosen A explicitly, do not correspond to a physical effect
on the ionization potential. They must rather be interpreted as arising from
that part of our particles whose energy state is really in the neighbourhood
of that corresponding to A. These particles can be thought as ionized or
free, according to the choice of A. This must result in a dependence on A
of the ratio between the numbers of those two kinds of particle states. We
are thus interested in an A-independent
correction on the Saha equation,
which we can find in the terms with ,!?e%c. Retaining only those terms, we
can calculate from (12) and (16) that
f = v(g)*
(1 +
@‘K).
(22)
1414
For g’ we introduce
R. DEKEYSER
a notation that corresponds
more to the usual formula-
tion of the Saha equation.
g’ = exp(--BEI)
where El is the lowest hydrogen
.ZB,
(23)
energy and ]Ell the undisturbed
ionization
energy, and Zg is the partition function for the bound states of an atom,
with the energies counted from the ground state. With these and (18), we
have
V
N”
(
s
>
’ (1 +
be%)
2nm
4
or
v
N”2
_
(4 8122
eBwll -Kea)ZB
NO
This result is in accordance with the calculations of E c ker and Krijllis)
for the effective ionization potential in a low density plasma.
3 5. Cut-independent
expression for the partition fzlnction; pressure. Our
(20)-(21), though not explicitly dependent on A, still contains N”,
which is a function of A. Until now, we always discerned between free and
bound particles, although there is no sharp physical discrimination between
them (a discussion of this fact has been given in reference 7). This way of
looking is also pronounced in the obtained expressions for the partition
function. It was our aim to derive results, independent of the definite choice
of the ionization criterion, at least in the first order in E. If this is true, it
must be possible to obtain an expression for the partition function, not
containing A at all.
result
We first note that the statistical
approximation formula, as
factor
can be written,
with Stirling’s
N!
N/t!
N”N-N”
(25)
=
n being defined as N”/N, the degree of ionization. This means that it
indicates the fraction of electrons having energy above our arbitrary cut.
An elementary calculation gives us
f
n=2Ng
Y
1+--- 4Ng
f
1
1 .
(26)
The A-dependent parts of both f and Ng are of order E in comparison with
the A-independent part of f. The A-independent part of Ng, which we called
PARTITION
Ng’, can be either
FUNCTION
small or large, depending
In either case, however,
of the form
we can obtain
n = no i- q
and PSObeing the solution,
and
OF A HYDROGEN
f by the A-independent
~
!7
a0
and density.
solution
for (26)
= O(&)
(27)
which is provided by (26) after substitution
expressions
of g
g’ and
’ (1 +
This solution
on temperature
an approximate
with
1415
PLASMA
#/%+c).
satisfies
92:
F’
l-9.20
(28)
-Ng”
The exact meaning of this no is not clear from its definition. It can perhaps
be interpreted as a kind of effective degree of ionization, and it certainly
does not differ much from the real degree of ionization, irrespective of the
way in which one defines it. It depends on A only through ,6e2K, which is
of order E. As the A-dependence of K is again an effect of order E, the total
A-dependence of no is of the second order in E, which we neglect.
We now calculate the N-th root of the product of the statistical
and
electronic part of the partition function (20) :
en-1
n
(N”g’ + F’) = Ng’ en’-’ eq
1+
Ng,cn:‘+
4) ) =
- W
1
_
eno’
no [l
+ Ok2)1*
It is worth noting that in the case of no m 1, this expression reduces to
F’/no. We thus derived the following expression for the partition function,
independent of the cut up to the second order in E:
Ng’
no-1
c-To--
With this, we can derive an expression
*=kT-&
.
for the pressure, from the formula
log 2.
This yields
P
kTN
1
=v+
2 1 -no
no
dna
dV*
1416
R. DEKEYSER
no depends on the volume only through F’. Derivation
where F’ depends
function
dno
fio(l -
dV
2 -
on V explicitly
1
dF’
F’
dV’
no)
no
and implicitly
of no. We can make an iteration
of (28) gives us
through
of these derivatives
K, which
is a
in powers of
E, and in the first order the result is
Wethen
1 dF’
__=_
1
F’dV
V
1’
have for the pressure
PV =
NkT
(1 + no) -
no
pe2K
3(2 - no)
(30)
We could try to obtain a similar expression by adding the ideal gas result
for the neutral particles, and the classical expression for the pressure of a
plasmais) and we would get
yT
= (1 -no)
With our method, we obtained
which for no m 0 reduces to half
tion can be understood from the
number of ionized particles will
+ 2fio[l
- $q.
a correction to the term containing /?e2K,
its classically expected value. This correcfact that by compression of the volume the
decrease by forming neutral bound states.
I should like to thank Professor I’?. G. van
Acknowledgements.
Kampen
for his hospitality at the University of Utrecht, for suggesting
the subject
Bouckaert
stimulating
Recei&d
and for his helpful advice. I am also indebted to Professor L. P.
and to Dr. D. Montgomery
for their kind interest and
discussions.
30-3-65
REFERENCES
1) Herzfeld,
K., Ann. Physik 51 (1916) 261.
2)
3)
Fowler,
Fermi,
R. H., Phil. Mag. 45 (1923)
E., Z. Phys. 26 (1924) 54.
4)
5)
6)
Becker,
Planck,
Riewe,
7)
R., Z. Phys. 16 (1923) 325.
M., Ann. Physik 75 (1924) 673.
K. H. and Rompe,
R., Z. Phys. 121 (1938) 79.
R. W., Comptes rendus de la VI e Conf6rence
Larenz,
d’Ionization
dans les Gas, Paris (1963),
Guernsey,
R. L., Phys. of Fluids 7 (1964) 792.
8)
1.
Tome
I, p. 219.
Internationale
sur les Ph&om&nes
PARTITION
FUNCTION
OF A HYDROGEN
1417
PLASMA
9) Shure, F., Phys. Rev. Letters 12 (1964) 353.
10) Harris,
G. M., Phys. Rev. 125 (1962) 1131 and 1331% (1964) 427.
11) Cf. Thompson,
W. B., “An introduction to plasma physics”, Pergamon Press, Oxford (1962),
p. 19 or Bates, D. R., “Atomic and molecular processes”, Academic Press, New York (1962),
p. 190.
12) Ecker, G. and Kriill, W., Phys. of Fluids 6 (1963) 62.
13) Cf. Balescu,
R., “Statistical Mechanics of charged Particles”, Interscience Publishers, New
York (1963), p. 249.
ERRATUM
The difference in the intermolecular
[Physica
potentials
of Ha and Dz
29 (1963) 13931
by A. K. BARUA
and A. SARAN
In this paper the stretching of a hydrogen molecule was calculated by
treating the rotator classically.
This, however, is not strictly valid for
temperatures
well below room temperature.
At these temperatures
the
following expression should be used.
C Z(Z +
1
2322pr,v2
dr=
z
1)(2Z +
s (21 +
1
1) 0 e-zcz+l)o’kT
1) ,-W+WW’
+
where ,Dis the reduced mass, v the frequency
internuclear
distance, h Planck’s constant,
3ha
___&+$,
1
eWkT _
1’
(l)
of vibration, ye the equilibrium
a the Morse constant
of the
molecule and o = h2/8n2pre2.
The calculated values of 6 by using eqn. (1) are shown in table I together
with the experimental values. The maximum difference between the present
calculated values of 6 and those given in the paper is about 2o/Oonly.
TABLE
T OK
&.l,
1
x lo*
6.X@ x 102
98
)
123
1 173
4.71
4.57
4.21
4.86
4.80
4.58
)
I
223
1 273
) 323
373
3.81
3.45
3.16
2.86
4.22
3.60
2.64
1.11
/
423
2.54
-0.74